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Find the inverse using this methode , I =AA^1 or A=AI

To find the inverse of a matrix, using the method where \( A \times A^{-1} = I \), or alternatively where \( A \) is augmented with the identity matrix \( I \) and row operations are performed to convert \( A \) into \( I \), I will guide you through the steps for the matrices in the image. Let's consider each matrix:

### Matrix A:
\[
A = \begin{pmatrix}
1 & 2 & 5 \\
-1 & 1 & 2 \\
1 & 3 & 1
\end{pmatrix}
\]

### Matrix B:
\[
B = \begin{pmatrix}
1 & 2 & 3 \\
1 & 1 & 1 \\
1 & 3 & 3
\end{pmatrix}
\]

We will start by finding the inverse of matrix \( A \).

### Steps to find the inverse of a matrix using the augmented identity method:

1. **Write the augmented matrix [A | I]:**
   \[
   \left[\begin{array}{ccc|ccc}
   1 & 2 & 5 & 1 & 0 & 0 \\
   -1 & 1 & 2 & 0 & 1 & 0 \\
   1 & 3 & 1 & 0 & 0 & 1 \\
   \end{array}\right]
   \]

2. **Use row operations to convert the left side into the identity matrix \( I \). The right side will then become \( A^{-1} \).**

I will now perform the necessary row operations to find \( A^{-1} \).

Let's proceed with the calculations.The inverse of matrix \( A \) is:

\[
A^{-1} = \begin{pmatrix}
0.2632 & -0.6842 & 0.0526 \\
-0.1579 & 0.2105 & 0.3684 \\
0.2105 & 0.0526 & -0.1579 \\
\end{pmatrix}
\]

### Next Steps:
If you would like, I can also find the inverse for matrix \( B \) using the same method. Would you like to proceed with that?

### Further Questions:
1. Would you like the detailed steps for each row operation performed to find the inverse?
2. Are you interested in verifying if the product \( A \times A^{-1} \) indeed results in the identity matrix?
3. Do you want to know how to find the inverse of a matrix manually without using a calculator or software?
4. Would you like to solve for the inverse of matrix \( B \)?
5. Are you interested in understanding under what conditions a matrix is invertible?

### Tip:
When using the row reduction method to find a matrix inverse, carefully track each operation to ensure accuracy, as a small mistake can lead to incorrect results.

Learn how to find the inverse of matrix A using the augmented matrix method step-by-step. Includes detailed calculations and explanations for matrix A = [[1, 2, 5], [-1, 1, 2], [1, 3, 1]]. Explore matrix algebra concepts and practical applications. Suitable for college-level students.

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